1 / 47

CHAPTER 2 POLYNOMIAL RATIONAL FUNCTIONS

2.1. COMPLEX NUMBERS. Objectives. Add

bronwen
Télécharger la présentation

CHAPTER 2 POLYNOMIAL RATIONAL FUNCTIONS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

    1. CHAPTER 2 POLYNOMIAL & RATIONAL FUNCTIONS

    2. 2.1 COMPLEX NUMBERS

    3. Objectives Add & subtract complex numbers Multiply complex numbers Divide complex numbers Perform operations with square roots of negative numbers

    4. i = the square root of negative 1 In the real number system, we cant take the square root of negatives, therefore the complex number system was created. Complex numbers are of the form, a+bi, where a=real part & bi=imaginary part If b=0, a+bi = a, therefore a real number (thus reals are a subset of complex #) If a=0, a+bi=bi, therefore an imaginary # (imaginary # are a subset of complex #)

    5. Adding & Subtracting Complex # Add real to real, add imaginary to imaginary (same for subtraction) Example: (6+7i) + (3-2i) (6+3) + (7i-2i) = 9+5i When subtracting, DONT FORGET to distribute the negative sign! (3+2i) (5 i) (3 5) + (2i (-i)) = -2 + 3i

    6. Multiplying complex # Treat as a binomial x binomial, BUT what is i*i? Its -1!! Why?? Lets consider i raised to the following powers:

    7. EXAMPLE

    8. Dividing Complex # It is not standard to have a complex # in a denominator. To eliminate it, multiply be a well-chosen one: ( conjugate/conjugate) The conjugate of a+bi=a-bi We use the following fact:

    9. EXAMPLE

    10. 2.2 Quadratic Functions Objectives Recognize characteristics of parabolas Graph parabolas Determine a quadratic functions minimum or maximum value. Solve problems involving a quadratic functions minimum or maximum value.

    11. Quadratic functions, f(x)= graph to be a parabola. The vertex of the parabolas is at (h,k) and a describes the steepness and direction of the parabola given

    12. Minimum (or maximum) function value for a quadratic occurs at the vertex. If equation is not in standard form, you may have to complete the square to determine the point (h,k). If parabola opens up, f(x) has a min., if it opens down, f(x) has a max. This parabola opens up with a steepness of 2 and the minimum is at (1,1). (graph on next page)

    13. Graph of

    14. 2.3 Polynomial Functions & Their Graphs Objectives Identify polynomial functions. Recognize characteristics of graphs of polynomials. Determine end behavior. Use factoring to find zeros of polynomials. Identify zeros & their multiplicities. Use Intermediate Value Theorem. Understand relationship between degree & turning points. Graph polynomial functions.

    15. The highest degree in the polynomial is the degree of the polynomial. The leading coefficient is the coefficient of the highest degreed term. Even-degreed polynomials have both ends opening up or opening down. Odd-degreed polynomials open up on one end and down on the other end. WHY? (plug in large values for x and see!!)

    16. Zeros of polynomials When f(x) crosses the x-axis. How can you find them? Let f(x)=0 and solve. Graph f(x) and see where it crosses the x-axis. What if f(x) just touches the x-axis, doesnt cross it, then turns back up (or down) again? This indicates f(x) did not change from pos. or neg. (or vice versa), the zero therefore exists from a square term (or some even power). We say this has a multiplicity of 2 (if squared) or 4 (if raised to the 4th power).

    17. Intermediate Value Theorem If f(x) is positive (above the x-axis) at some point and f(x) is negative (below the x-axis) at another point, f (x) = 0 (on the x-axis) at some point between those 2 pts. True for any polynomial.

    18. Turning points of a polynomial If a polynomial is of degree n, then it has at most n-1 turning points. Graph changes direction at a turning point.

    19. Graph

    20. Graph, state zeros & end behavior END behavior: 3rd degree equation and the leading coefficient is negative, so if x is a negative number such as -1000, f(x) would be the negative of a negative number, which is positive! (f(x) goes UP as you move to the left.) and if x is a large positive number such as 1000, f(x) would be the negative of a large positive number (f(x) goes DOWN as you move to the right.) ZEROS: x = 0, x = 3 of multiplicity 2 Graph on next page

    21. Graph f(x)

    22. Which function could possibly coincide with this graph?

More Related